What a space group extinguishes: the integral, zonal and serial reflection conditions of any of the 230 groups, in any of the settings the International Tables list. Ask about one reflection and the page names the symmetry operation that removes it.
You supply
A space group number or symbol. Non-standard settings are included, so P21/n, Pbnm and I2/a answer for themselves rather than being redirected to a sibling with different conditions. A reflection is optional.
Reading it
Absences narrow the space group and rarely determine it: the 230 groups produce only 80 distinct patterns of absence, and 37 of them extinguish nothing at all. When your conditions fit more than one group, the page names the others.
Conditions belong to a setting, not to a space group number. Pnma, Pbnm and Pmcn are one space group with its axes labelled three ways, and the three tables differ: the mirror that empties hk0 in one of them empties 0kl in another. Every setting in the International Tables is here with its own operations, so P21/n and P21/a answer for themselves.
Point at any element for its type, its direction and where it sits. Axes are
drawn thicker the higher their order; a screw axis or a glide plane is dashed, because its
operation carries a translation the pure one does not. Switch a class off to see through a
dense group.
The cell drawn here is a representative one: its shape obeys every constraint this group's symmetry imposes and nothing else, because a space group fixes the equalities among the cell constants and never their values. Read the right angles, the equal axes and the 120° where they appear; do not read the axial ratios.
The crystal class, projected
Every space group leaves a crystal class behind when its translations are
taken away, and this is mmm — the
8 operations of
Pccn seen as directions rather than as
places. On the left, one general direction and everywhere the class sends it; on the right,
the same class written as its axes and mirrors.
Both discs are the same sphere seen from the same place: every direction is joined to the far pole and marked where that line crosses the equator, so the centre of the disc is straight up c, the rim is ninety degrees away from it, and the angle between any two directions survives the projection. a points down the page and b across it.
A filled dot is a direction in the upper half of the sphere and an open circle one in the lower; where the group has a mirror across the page the two coincide, and the mark is a dot inside a circle. A comma means the operation that produced it turned the object inside out.
Point at any pole or any symbol to have it name itself.
8 poles, drawn as 4 marks.mmm has a mirror across the page, so each pole in the upper half of the sphere lands exactly on top of one in the lower half. Every one of those pairs is drawn as the Tables draw it — a dot inside a circle — and pointing at it names both.
4 of the 8 poles carry a
comma. Those images are of the opposite hand, produced by an operation that turns
the object inside out — a mirror, an inversion, or an inversion axis. A structure
built from a single enantiomer cannot have this symmetry, because the group would demand
the other one alongside it.
The International Tables turn the page for monoclinic groups and draw the unique axis in the plane of the paper. This projects along c whatever the group is, so that all three diagrams on this page share one orientation — in a group with the unique axis b, the two-fold therefore lies across the disc instead of standing at its centre.
General position
The 8 symmetry operations of
Pccn, written as coordinate triplets. Each one
says where the group sends a point at x, y, z.
x,y,z
-x,-y,-z
-x+1/2,-y+1/2,z
x+1/2,y+1/2,-z
x+1/2,-y,-z+1/2
-x+1/2,y,z+1/2
-x,y+1/2,-z+1/2
x,-y+1/2,z+1/2
Where those points are
drag to rotate · scroll to zoom
Look along
0.13, 0.21, 0.31 is a general position. No operation of Pccn leaves it where it is, so the group sends it to 8 distinct points — one per operation, which is what makes the multiplicity equal to the order of the group.
That is the general position the drawing opens on. Type a point into the
form to move it — the picture and the multiplicity follow.
A filled circle is the point as you gave it; a hollow one is a mirror image of it, produced by an operation of the second kind — an inversion, a mirror, a glide or a rotoinversion. The International Tables mark the same distinction with a comma inside the circle. It matters for anything chiral: those copies are not superimposable on the one you typed.
The 8 points
#
x
y
z
From
1
0.13
0.21
0.31
x,y,z
2
0.87
0.79
0.69
-x,-y,-z
3
0.37
0.29
0.31
-x+1/2,-y+1/2,z
4
0.63
0.71
0.69
x+1/2,y+1/2,-z
5
0.63
0.79
0.19
x+1/2,-y,-z+1/2
6
0.37
0.21
0.81
-x+1/2,y,z+1/2
7
0.87
0.71
0.19
-x,y+1/2,-z+1/2
8
0.13
0.29
0.81
x,-y+1/2,z+1/2
Reflection conditions
Class
Condition
From
0kl
l = 2n
zonal
h0l
l = 2n
zonal
hk0
h + k = 2n
zonal
h00
h = 2n
serial
0k0
k = 2n
serial
00l
l = 2n
serial
Reading down the table: integral conditions come from the lattice centring and apply to every hkl; zonal conditions come from glide planes and empty one plane of reciprocal space; serial conditions come from screw axes and empty one row. The same three kinds are named under the stick pattern on the HKL page, which applies the integral ones only.
A class in this table stands for its own zone and every zone the point group carries it onto. In P63/mmc the row for hhl also governs (−2, 1, l) and (1, −2, l); in a cubic group the row for hhl also governs hkh and hkk. Seventeen of the settings here extinguish reflections that no literal reading of a class label covers, which is why the check below asks the symmetry operations rather than this table.
Is a reflection there?
Put three Miller indices into the form and this says whether
Pccn allows that reflection, and which
symmetry operation removes it if it does not.
What the absences do not tell you
These same conditions are produced by
one other space group.
Systematic absences cannot separate them, and no amount of care with the data will change
that: the choice between them is made on intensity statistics, on the structure solving, or
on chemistry.
Moving the origin multiplies every structure factor by a phase and changes no intensity, so the two origin choices the tables give for 24 space groups have identical conditions. Only one is listed here.